Store-bought pie crust can be substituted for the homemade version here for a little less work. Slideshow: More Chicken Recipes
In a bowl, blend together the flour, butter and 1 teaspoon salt with your fingertips or a pastry blender until the butter is mostly combined with some pea-sized lumps. Using a fork, stir in 5 tablespoons of the water. Squeeze a handful of the dough, if it is still crumbly, stir in the remaining water, 1 tablespoon at a time. Gather the dough into a ball, then form it into 2 disks and wrap each disk in plastic wrap. Chill the dough at least 1 hour.
On a floured work surface with a floured rolling pin. roll the dough out into 2 11-inch rounds. Place 1 round of dough into a 9-inch pie place and rim the dough so that there is 1-inch of overhang. Chill the dough until ready to use. Preheat the oven to 400°F.
In a large cast-iron or heavy skillet, heat the oil over medium high heat until hot, then stir in the onion, garlic, carrots, potato, 3/4 teaspoon salt and 1/2 tea-spoon pepper. Cook, stirring occasionally, until golden, 8 to 10 minutes. Stir in the flour and cook, until incorporated, about 2 minutes. Pour in the stock and bring to a boil, then stir in the chicken. Season the filling with salt and pepper to taste.
Let the filling cool to warm then place the filling in the pie shell. Place the remaining round of pastry dough over the filling and trim and crimp the edge. Brush the top crust with the egg mixture, then cut steam vents in the top crust. Bake until the filling is bubbling and the pastry is golden, 40 to 45 minutes.